I want to unite every piece of this into a connected unified mesh without anything to clean up afterwards
A possible way is using the Shrinkwrap Modifier, though it might not affect the look, it will most likely radically change your mesh geometry.

First we have an object consisting of multiple meshes:



Then we make a high resolution (in terms of vertices) sphere around it:



Then we put a Shrinkwrap Modifier on it. Choose the initial object as your target. Choose "Project" as the Mode. And check the box "Negative". Lastly click the Apply button.



I have no idea which is which.



Seems the icosphere was more then enough high rez. You might want to tone it down a little depending on the level of detail of the initial mesh.